Neural network as (BDI) agent - running in continuous mode (that do inference in parallel with learning)?
You might be interested in the Clarion cognitive architecture, developed by Prof. Ron Sun and collaborators. 

Full disclosure: I am a student in Ron Sun's Cognitive Architecture Lab.

Brief Description of Clarion

Clarion agents are composed of several subsystems, each of which may contain several neural networks. These subsystems include the Action-Centered Subsystem (ACS), the Non-Action-Centered Subsytem. The ACS controls action decision making, the NACS stores general knowledge. During an activation cycle, the ACS might request information from the NACS, setting the NACS activation cycle in motion. 

Strictly speaking, Clarion does not adopt the BDI framework. But, various components of a Clarion agent can be put in correspondence with BDI concepts. BDI beliefs correspond roughly to knowledge in the ACS and NACS. The Motivational Subsystem (MS) contains agent drives and sets agent goals, which roughly correspond to BDI desires and intentions respectively. 

Clarion agents do not directly control when learning happens, but may control which subsystems are active based on task demands, and may output mental actions as part of their processing. Learning generally happens at the end of a subsystem activation cycle based on feedback from the environment or from the agent's own motivational and metacognitive subsystems.

Links

A page on Prof. Ron Sun's site links to several resources on Clarion. 

Currently the most up-to-date reference on Clarion is the book Anatomy of the Mind. A precis is available in the journal Cognitive Computation.